Cutoff effect in molecular dynamics simulations of interaction induced light scattering spectra

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ELSEVIER Computer Physics Communications 105 (1997) 151-158

Computer Physics Communications

Cutoff effect in molecular dynamics simulations of interaction induced light scattering spectra

Victor Teboul 1, St6phane Chaussedent

Laboratoire des Propridtds Optiques des Matdriaux et Applications, Universitd d'Angers, Facultd des Sciences, 2 boulevard Lavoisier, 49045 Angers, France

Received 22 November 1996

Abstract

A strong dependence of the interaction induced light scattering integrated intensity with the cutoff radius was previously reported in noble liquids molecular dynamics simulations (A.J.C. Ladd, T.A. Litovitz, J.H.R. Clarke, L.V. Woodcock, J.

Chem. Phys. 72 (1980) 1759; and J.H.R. Clarke, L.V. Woodcock, Chem. Phys. Lett. 78 (1981) 121). It was concluded that the uncertainties arising from this effect were much larger than can be tolerated in any evaluation of the light scattering mechanism. No real explanation or solution ever appeared in the literature on this problem. Here we report a very strong dependence of the interaction induced spectra calculated by molecular dynamics simulations on the number of atoms used in a liquid argon simulation. We show that this effect is proportional to the square of the density involved in the simulation

- - 4

and that it obeys to a Rcutoff power law with the cutoff radius Rcutoff. To obtain accurate spectra up to high frequency shifts, large cutoff radii have then to be used corresponding to thousands of atoms in the simulation. The time of the simulation is then prohibited. An explanation for both effects is proposed and two different methods to solve this problem are tested.

@ 1997 Elsevier Science B.V.

PACS: 34.50.-s; 33.20.Fb; 33.70.-w

Keywords: Light scattering; Interaction induced; Polarizability; Size effect; Molecular dynamics

1. I n t r o d u c t i o n

Interaction induced light scattering (IILS) is the name for the Raman spectroscopy which arises from the intersction-induced variations o f the polarizability of a sample. IlLS is now a well-established experimental method to obtain information on intermolecular interactions [ 1 - 8 ] . At low densities, binary interactions are predominant and exact calculations o f the induced spectra are possible provided that an intermolecular potential and an interaction polarizability

l E-mail: Teboul@univ-angers.Fr

model have been chosen [9-11 ]. When higher densities are involved, three and four body interactions be- come important and molecular dynamics simulations have, among other methods [ 1 2 - 1 5 ] , to be used in order to provide spectral intensities [ 16-21] or mo- ments [ 15,22,23] using the intermolecular potential and polarizability as input in the calculation. In the low frequency range, scattered intensities are mainly due to long distance interactions, which are well described by the simple Dipole Induced-Dipole ( D I D ) polarizability model [24] for noble gases. For higher frequency shifts, shorter range interactions have to be taken into account in the polarizability model to re-

PII S0010-4655(97)00079-9

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152 V. Teboul, S. Chaussedent/Contputer Physics Communications 105 (1997) 151-158 produce correctly the experimental spectra. This has

been demonstrated in the case of noble gases in the low density limit [9,10], but was never seen in noble liquids probably due to the limitation of the frequency shifts investigated. As IILS experimental spectra are obtained with higher and higher accuracy, higher frequency shifts are investigated and the question of ob- taining high frequency spectra from the theoretical point of view becomes of some importance. Molecu- lar dynamics simulation seems then to be the simplest method to study the density behavior of interaction induced spectra. However, some attention must be paid to the limitations of the simulation method, because of the systematic errors which are intrinsic to the system under investigation [25,26].

Due to the finite number N of atoms used in the simulation, a cutoff radius is introduced in the calculation of the correlation function, in order to conserve spher- ical symmetry. This cutoff radius has to be smaller than half the length of the box containing the N atoms at the density used in the simulation. A few years ago A.J.C. Ladd, T.A. Litovitz, J. H.R. Clarke and L. V.

Woodcock have found a strong dependence on the integrated intensity (zero momentum) o f the simulated spectra of liquid argon, arising from the cutoff radius used in the calculations [21,27]. They found that the error arising from truncation was many times larger than the total experimental intensity for liquid argon.

As a consequence they concluded that all discrepan- cies between experimental and computed integrated intensities were within the uncertainties due to com- putational artifacts. They did not find strong effects on the spectrum shape [21 ].

We report here a strong dependence of the simulated spectrum arising in liquid argon at higher frequency shifts than those displayed in their calculations [21 ].

To obtain spectra up to high frequency shifts, large cutoff radii have then to be used, leading to a large number N~o.f atoms in the simulation. Since the time of the simulation is increasing roughly-as N 2, this method will be quite time consuming. To solve this problem, we propose two different methods. The first one considers the interactions coming from the images of the atoms in thesimulation box. The second method uses a smooth cutting edge in the correlation function calculation. In our calculations we chose argon for its relatively heavy mass, and the temperature and density (130 Kelvin and 615 amagat) were chosen due to

experimental considerations.

Our paper is organized as follows. Section 2 de- scribes the procedure used for the simulations. The calculated spectra and their comparison are then dis- cussed in Section 3, and the methods proposed to eliminate the cutoff effect in the interaction induced scattering spectra are also described and tested in this section. Section 4 is then the conclusion.

2. Details on the calculations

The simulations were carried out on a Hewlett Packard HP 715/50 workstation and partly on a Cray YMP computer. We have used usual molecular dynamics simulations with a Verlet algorithm [28]

to solve the equations of motion with a time step of 10-14seconds. The Lennard-Jones 6-12 potential for liquid argon ULj =

4e((tr/r)12_

( t r / r ) 6 ) (with

e/ko

= 120 K and tr = 3.4/~, where ko is the Boltz- mann constant) was used for simplicity. However, in preceding simulations Vermesse et al. have shown that the intermolecular potential has only small effects on interaction induced light scattering line shapes of liquid argon [29]. As usual, the simulations begin with a box o f crystalline symmetry argon which is warmed up to the liquid or gas phase. Accordingly we used a number of atoms in a cubic box equal to 4n 3, where n is an integer number. Various sample sizes were used to reveal the size (or cutoff) effect on the simulations. They varied from 108 to 864 atoms.

The equilibrium configurational averages have been performed at up to 106 time steps. We assumed as usual [30] that the intermolecular potential and the intermolecular polarizability are pairwise-additive.

The correlation function in the classical approximation is then given by [ 15,18,31]

CXY(t)

= (normalization factor) • - - I M

M

• + , ) , (1)

tn=l i,j,k,I

where At is,the time step ( 10 - 1 4 seconds in our case).

We define here the cutoff radius Rcutoff as the maximum range of atoms i and j , and atoms k and l which contribute in the summation via the terms T/) (mAt) xy and

T~Y(mAt + t).

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V. Teboul, S. Chaussedent/Computer Physics Communications 105 (1997) 151-158

153

Tq is given in the general case by

~ (a)_ (b))

~xq xq

7/~ (")xa') =

OtSab -- l f l 6a b

r7

,

(2)

where i and j are the atom labels and x}j a) (a = 1,2 or 3) are the coordinates

x O, yq

and

zq,

respectively, of the relative position

rij

between atoms i and j. Quan- tities

a(rij(t))

and

fl(rq(t))

stand for the trace and the anisotropic part respectively of the interaction induced polarizability tensor [ 1 ]. In our calculation we used the first order Dipole Induced-Dipole (DID) polarizability [24,1 ]

We can then obtain the spectral intensity from the Fourier transform of the correlation function. As

C(t) = C ( - t )

we may write the Fourier transform as

o o

In(o)) = • [cos(tot)C(t) dt.

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7 r j o

To insure that no spurious effects are coming from the Fourier transformation process, different numerical Fourier transform methods were tested [28,32].

To obtain the intensity, we write

l ( p ) = kok3sln ( p )

e hcV/2knr , (8)

6ag (3)

a = 0 and f l = r3 ,

where a0 is the polarizability of an isolated atom, which is 1.68 A 3 for argon. In this case, we have

7~jX=o~--~ 1 - r 2 ) = r-ffq I--r--~ij] , xv ( x i j Y i J ~ = 6 a ~ ( x q _ _ _ _ ~

( 4 ,

T i ) = f l \--~q ] \ rij

] "

and symmetric relations for the other coordinates.

The term in parenthesis in Eq. ( l ) is a normalization factor. We chose this term equal to

Vcell/N 2

(where

Vcell

is the volume of the simulation cell) in order to obtain an intensity independent of both the number of atoms in the simulation, and the density within the low density limit.

To increase the configurational average we then add the correlation functions calculated for different ge- ometries. The depolarized geometry correlation function can then be written as

C(t) = ~{3[ CXX(t) + CY>'(d) + CZZ(t) ]*

+[CXy(t) +CXZ(t) -FC>'Z(t)]}. (5)

The stars mean here that the isotropic part ot of the polarizability was set to zero for the calculation of

CXX( t), CYV ( t)and C zz ( t).

In this case the different correlation functions are theoretically related by

3CXX(t )*

= 3 C > ' Y * ( t ) =

3CZZ(t )*

=CXy(t) = C xz(t) = C yz(t). (6)

where the exponential term is introduced for detailed balance correction and k0, ks are the wave numbers of the incident and scattered light, respectively. We chose a typical laser wavelength of 5145/~. The values obtained in this way for the intensities are given in cm 6, a commonly used unit for low density interaction induced light scattering spectra.

3. Results and discussion

The integrated intensity dependence on the cutoff radius is shown in Fig. 1. We find the oscillating dependence described by Ladd et al. [21,30.]. Moreover, our results roughly agree in shape and values with their model [21]. We think that if previous simulations did not agree in intensity [21] with their theoretical model, it was probably due to limitations in the time steps used in the simulations. We used here a very high number of time steps, one million instead of five thousands in their simulations [ 21 ]. The oscillations of the integrated intensity displayed in the figure correspond exactly to the oscillations of the

g(r)

function [ 30] which represents the probability to find an atom at a distance r from another one. This as- sumption can easily be verified in Fig. 1 since the

g(r)

function is maximum for integer numbers of the potential parameter tr (3.4 /~) and minimum in between. In other words, the integrated intensity oscil- lates with the number of atoms at the limit of the cutoff radius. Due to that dependence there is an uncertainty on the integrated intensity calculated with molecular dynamics simulations [21,27,30]. Fig. 1 shows that this uncertainty is rather weak in our case (less than

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154 V. Teboul, S. Chaussedent/Computer Physics Communications 105 (1997) 151-158

0.8 I

0

"~ 4"

l:l 0.6 -- l J

" ~ I 11

9 ~ t~

- - II

0.4

0.2 J

0

a l i l ~ l a l ~ l

I I I 14.1 ,~

I I I l 1 ~ . . .

1111.1 -'I~ "P - " ~ ~ - - G 0 II I ~ n

I1"~" +E~ ~ A

11

I I I I I I I I I I I

10 20 30 40 5 0 60 70

c u t o f f r a d i u s / A n g s t r o m

Fig. I. Integrated intensity (zero momentum) as a function of the cutoff radius for argon at a temperature of 130 K and a density of 615 amagat. The crosses represent direct simulations results with 256, 500 or 864 atoms, the squares are for simulations using the images method and the triangle represents a simulation with 256 atoms and a smooth cutoff. The dashed line is the result of the theoretical model o f Ladd et al. [21] in the same units (,~9).

There is a factor 15/2 between our ~9 units and most o f the low density .~9 units previously used for the integrated intensity. The values can be transformed in cm 5 (units o f the integral of the spectrum) by multiplying by the factor k 4.

8 percent) if one considers more than 256 atoms in the simulation or more precisely for a cutoff radius higher than 10/~. We show in Fig. 2 liquid argon spectra calculated at a temperature of 130 K and a density of 615 amarat with different ct~toff radii. The spectra are the same in absolute values and shapes up to a limit frequency shift where a plateau is reached. The spectra seem then to be the superposition of the real spectrum and a slowly decreasing spectrum which intensity but not shape depends on the cutoffradius. The detailed balance correction however enhances slightly the displayed effect by increasing the high frequency shift intensities. We have tested that the effect is not due to the boundary conditions at the limit of the box

1 0 7 i I J I ~ I '

10 e

9 r 1 0 s

o Q 1 0 4

10 ~

.~ 1 0 2

_=

10 ~

A r g o n 6 1 5 a m a g a t 1 3 0 K

a

%.

10 ~ I 1 I ] I I f

0 100 2 0 0 3 0 0 4 0 0

1/ / c m - I

Fig. 2. Frequency spectra ( p = 615 amagat and T = 130 K) for different cutoff radius Rcuto,. (a) Simulation with 108 atoms corresponding to Rcutotr = 9.3 ,~. (b) 256 atoms corresponding to Rcutofr = 12.5 ~. (c) 500 atoms corresponding to Rcutofr = 15.6 /~. (d) 864 atoms corresponding to Rcutofr = 18.7 ~ . (e) 256 atoms and their first images corresponding to Rcutofr ='37.4/L The statistical uncertainty corresponds to the dispersion o f the points.

since it is not directly dependent on the number of atoms but on the cutoff radius, different simulations with the same cutoff radius and different numbers of atoms leading to the same spectrum. When a shorter cutoff radius is used, the low frequency intensities de- crease and the spectrum tends to a roughly Constant curve. The very beginning of this effect can be seen on curve (a) of Fig. 2 which corresponds to a 108 atoms simulation. This can mean that the long distance part of the interaction polarizability leads to the predominant contribution at "high frequency" shifts.

This conclusion is in contradiction with the idea that long distance means slow variations (or long times) which in turn lead to low frequency shifts when the Fourier transform of the correlation function is calculated, and that in an analogous way high frequency shifts mean short distances and energetic interactions.

This apparent contradiction can be explained if the

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V. Teboul, S. Chaussedent/Computer Physics Communications 105 (1997) 151-158 155

9 10 4

,,-I

9 10 ~

9 , ~ 10 =

I~ 101

10 n a I'~ I I I I i a I

% -I-

%

%\

%

o % %

%

I I I I~tl I I

1 0 ~

1 0 0 , n I I I I I I J

10 1 0 '

c u t o f f r a d i u s / A n g s t r o m

Fig. 3. Intensity at a frequency shift of 400 cm - I as a function of the cutoff radius (+) in a Logarithmic/Logarithmic scale. On the last point, the interpolated value of the real spectrum has been subtracted (o) or not (+). The dashed line corresponds to a pure R -4 power law. _r

high frequency slowly decreasing spectrum does not arise from the neglected long range contributions but rather from the cross terms in the correlation function with one atom near the cutoff limit. Because when the cutoff is introduced, the number o f atoms inside the calculation is no longer constant, atoms crossing or not the frontier introduce a parasitic noise in the correlation function with a characteristic fluctuation time of the order o f the time step (At = 10 -n4 seconds) which can lead to this roughly fiat spectrum. This explanation can be tested with the dependence o f the high frequency intensities on the cutoff radius. In Fig. 3, we have plotted the 400 cm -n intensities for different cutoff radii in a logarithmic/logarithmic scale. A power law dependence is then a straight line in the figure. The dashed line representing a pure R~t4off dependence agrees remarkably well with the simulation results. I f the effect arises from atoms crossing the cutoff surface we expect that it would be proportional

%-

b

o,,.~

m ,'el

"7"

v

2 4 . 0

2 0 . 0

16.0

12.0

8 . 0

4 . 0

' I ' I ' I = ,1

i # i I e a i # t I

s ~' I I 9 1 4 9

/ i I

' t ~' #e

I

# t

/ ,'

ee u

- - u n

e

. 2

w

~ i i i i

P A 9

r I , +"

s ~

,.+

t J I

0 100

s I

a I ' i

o

o.o I , I , I , I ~ I ,

200 300 400 500 600 700

d e n s i t y / a m a g a t

Fig. 4. Intensityxdensity as a function of density for different frequency shifts. Squares correspond to the intensity at a frequency shift of 15 cm - t , triangles at 100 cm - t , and crosses at 400 cm - I in the cutoff effect region. Intensity xdensity in cm 6 xamagat have been multiplied by 1052 at 15 cm -I (squares), 1054 at 100 cm - t (triangles), and 1056 at 400 cm - I (crosses). The lines are fits of the points.

to the number of atoms crossing this surface, and then to the surface o f the sphere. If instead the effect arises from long range interactions neglected in the calculations, we expect that it would be proportional to the number o f atoms neglected which would lead, in the DID approximation, to an Rcutoff dependence with the -3 cutoff radius. An Rcuto ff dependence in the D I D ap- -4 proximation (f12 proportional to r -6 ( 3 ) ) then shows ( I ) that the effect is proportional to the surface o f the cutoff and that the first explanation is the right one. Finally, Fig. 4 shows the density dependence o f the simulated intensity for different frequency shifts.

Straight lines on the figure represent square density dependence and other shapes correspond to a dependence with higher powers o f the density. The typical dependence at the two frequency shifts 15 c m - t and 100 cm - t are just shown here for comparison with the

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156 E TebouL S. Chaussedent/Computer Physics Communications 105 (1997) 151-158 cutoffintensity (400 cm - l ) dependence with density.

At low frequency shifts ( 15 cm - I ) we find the usual many body contributions and at high frequency shifts, here 100 cm - t , the intensity behaves as the square of the density showing that binary interactions are there predominant. At 400 cm - t , the calculated intensity is almost purely induced by the cutoff effect. In this region we see in the figure that the cutoff intensity behaves as the square of the density, showing that the cutoff effect is mostly induced by binary interactions.

This justifies the definition we used for the cutoff radius since there is no ambiguity for binary interactions in which atoms to be used for the minimum cutoff distance.

We then have tested two different methods to eliminate the cutoff effect on the high frequency part of the simulated spectra. We will now describe and test these two methods. In usual molecular dynamics calculations one considers a set of N particles located in a cubic box. To be able to suppress boundary effects and to allow for a connection to real world systems one then usually considers the system to be embedded in a large three-dimensional array of copies o f the box. As in the Ewald method [28,33] we have calculated here the interactions of the atoms of our box with the array of the outside copies to obtain the effect of long range interactions on the spectrum. In the correlation function calculation (1) the summation is then enlarged in this method to images of the atoms, located outside the .central simulation box. The maximum cutoff radius which can be used in the simulation, for a given number N of atoms, is then increased. This method is relatively simple and rapid since the positions of the images have not to be calculated again in the simulation. As our method introduces atoms which are cor- related with some atoms of the central box we have to test that no spurious effect is introduced in this way. To test the method several simulations were performed, with and wi~out interadtions with the copies of the central box. We display some of those tests in Fig. 5.

For the sake of clarity we have smoothed the curves and eliminated 5 points on 6 in the result. We show in Fig. 5 the results of a simulation with 256 atoms and a cutoff radius of 12.5/~, and 108 atoms taking into account adjacent boxes, with the same cutoff radius. The two curves are identical. On the same figure the results of simulations of 500 atoms with a cutoff radius of 15.6 ./k and 256 atoms with adjacent boxes,

1 0 7 ' I ~ I u l

10 6

9 10 s

0 10 4

10 ~

~0 ~

-~1 Argon

~ 615 amagat

130 K S

a'= ~cr~ ^a ^_

~ ~ ~ b

c

1 0 0 I I I I , I t

0 I 0 0 2 0 0 3 0 0 4 0 0

v / c m -i

Fig. 5. Frequency spectra (p = 615 amagat and T = 130 K) for different cutoffmdii Rcutoff. (a) Rcutofr = 12.5 1~. (b) Re=oft = 15.6 /~.. (e) Rcutoff = 18,7 ,/k. The dashed lines correspond to direct simulations with respectively 256, 500 and 864 atoms and the squares to simulations using the images method (with respectively 108 atoms and first images and 256 atoms and first images). In these curves we used the same cutoff as in the direct simulations.

and 864 atoms with a cutoff radius o f 18.7/~ and 256 atoms with adjacent boxes. In each case the curves are identical within the statistical uncertainties. However, if the cutoff high frequency effect arises from the noise in the simulation introduced by the atoms coming in and out from the cutoff sphere, a much simpler way to eliminate it would be to introduce a smooth cutting edge in the correlation function calculation. We then tried this method with different kinds and sizes of windowing,functions. We found two kinds o f results: no modification at all of the calculated spectrum when the cutting edge was too sharp, and the decreasing spectrum for smooth enough cutting edges. We finally used the following windowing function:

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E TebouI, S. Chaussedent/Computer Physics Communications 105 (1997) 151-158 1 ( Ir( r - Rb )

~ ^for

r < Rb,

c~ k,2(--~M~---Rb)J

^forRb < r < RMa~x,

0 for r > RMax,

with RMax = L / 2 , where L is the length o f the simulation box, and Rb = RMax-Rw, where Rw = 0.8 A is the size of the cutting edge. But this function may have to be changed for different thermodynamical conditions since it must be connected to the mean velocity o f the atoms. The high frequency spectrum was there- fore also decreasing with this method, leading to the same limit spectrum than with the first method up to 400 cm - I , which is t h e limit o f the precision of our simulations if the Dipole Induced-Dipole polarizability is used.

4, C o n c l u s i o n

We have found a strong dependence of the interaction induced scattering spectra on the cutoff radius used in molecular dynamics simulations. This effect mostly affects the high frequency part o f the spectra, but also the integrated intensity o f the spectrum. The high frequency spectrum and the integrated intensity o f the spectrum must then be corrected from this effect before any comparisons with experimental results.

Moreover, since the effect shows a square density dependence and dominates the calculated spectra at high frequency shifts, the density dependence o f spectra at high frequency shifts not corrected from this effect must be taken with caution. An evaluation o f the mag- nitude o f the effect can be obtained directly from our curves. Fig. 3 gives for instance the cutoff radius dependence o f the intensity. Although our results have been obtained in the case o f liquid argon at a density o f 615 amagat and a temperature o f 130 Kelvin, they

#

can be obtained at other densities, using the density dependence (Fig. 4) o f the intensity o f the effect. We have shown that it was proportional to the square o f the density which is equivalent to no dependence in cm 6 units. The cutoff intensities can also be evaluated for other rare gases or at other temperatures, using the corresponding state law as our calculations make only use o f classical mechanics, We have then proposed two methods to eliminate the cutoff effect in molecular dynamics simulations. These methods were then

157 tested successfully, both eliminating the effect up to the precision o f our calculations.

A c k n o w l e d g e m e n t s

We would like to thank Andr6 Monteii for inter- esting discussions and also the Orsay University for providing us their computing facilities.

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Cutoff effect in molecular dynamics simulations of interaction induced light scattering spectra

Computer Physics Communications